Compressive Sensing
Compressive Sensing (CS) uses fewer linear measurements than implied by the dimensionality of an acquired signal. To reduce the acquisition rate, CS reconstruction methods exploit the structure of signals. To capture the structure, the most commonly used signal model is sparsity.
As used herein, sparse is a well known term of art in signal processing, and not a relative indefinite term. A “sparse” approximation estimates a sparse vector, i.e., a vector with most coefficients equal or approximately equal to zero, satisfying a linear system of equations given high-dimensional measured data and an acquisition matrix.
A CS-based acquisition system can be model asr=A(s)  (1)where A(·) is a linear function, s is in some appropriate signal space, and r is in a measurement space. The measurement space has a much lower dimension than the signal space. A number of possible properties of A(·), such as low coherence, the restricted isometry property, depending on the model, guarantee that the reconstruction is possible using an appropriate method.
CS is successfully applied in many imaging systems, in which, typically, the signal s to be acquired is a 2D image in Nx×Ny.
Using compressive sensing approaches, it has been shown that images can be acquired with measurements as few as 10% of the number of pixels NxNy. These gains are not as relevant in conventional visible-light imaging, where charge-coupled devices and metal-oxide-semiconductor sensor technology make measurements extremely inexpensive, which has had a significant impact in other modalities, such as medical imaging, low-light imaging, hyperspectral imaging, and depth sensing. However, the gains are significant for imaging systems that depend on complex costly sensors, such as “time-of-flight” (TOF) light or radar sensors.
Model-Based Compressive Sensing
A model-based CS framework provides a general approach to developing a large number of signal models and characterizing their suitability for CS acquisition. Models under this framework are created by imposing restrictions on the signal support. A fundamental operation is the projection of a general signal to the set of signals that satisfy support contraints on the model. As long as such a projection can be computed, common greedy CS reconstruction methods, such as Compressive SAmpling Matching Pursuit (CoSaMP) and Iterative Hard
Thresholding (IHT), can be modified to reconstruct signals in the model. Furthermore, it has been shown that a pair of approximate projections with different approximation properties is sufficient to guarantee accurate reconstruction, instead of an exact projection, see Hegde et al., “Approximation-tolerant model-based compressive sensing,” Proc. ACM Symposium on Discrete Algorithms (SODA). SIAM, pp. 1544-1561, January 2014.
One signal model, approximation tolerant model-based compressive sensing, is motivated by signals measured using 2D seismic imaging. A signal sεN×T according to that model is a matrix with N rows and T columns. Each row of the signal only has S non-zero entries, which are spatially close to the S non-zero entries of the row above or below. This is enforced by restricting an earth-mover distance (EMD) between the support of subsequent rows of the signal. The projection under the approximation tolerant model is performed by solving a sequence of network flow problems.
Optical Sensing
Optical sensing measures distances to a scene by illuminating the scene with optical signal and analyzing the reflected light. One optical sensing technique, e.g., light radar (Lidar), can be used with applications such as geoscience, remote sensing, airborne laser swath mapping (ALSM), laser altimetry, contour mapping, and vehicle navigation.
Conventional high-resolution, high frame-rate optical based systems typically use an expensive array of precision TOF sensors and illuminate the scene with singular pulses transmitted by a stationary laser. Alternatively, at the expense of reduced frame-rate, the laser scans the scene. A smaller sensor array, which can also scan the scene, acquires the reflection. The resulting system significantly lowers the cost, but requires the use of mechanical components, which can be prone to failure and cumbersome in mobile applications.
Compressive sensing can exploit significant gains in computational power due to the reduced sensing cost, and allow elaborate signal models and reconstruction methods, which, in turn, enable reduced sensor complexity. For example, some compressive depth sensing systems use a single sensor combined with a spatial light modulator and multiple pulses illuminating the scene.
A spatial light modulator is used to implement a variable coded aperture that changes with every pulse. However, that restricts those system to static scenes, see Howland et al., “Compressive Sensing Lidar for 3D imaging,” CLEO: Science and Innovations. Optical Society of America, 2011, CMG3, Kirmani et al., “Exploiting sparsity in time-of-flight range acquisition using a single time-resolved sensor,” Opt. Express, vol. 19, no. 22, pp. 21485-21507, October 2011, and Kirmani et al., “CoDAC: a compressive depth acquisition camera framework,” IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2012, pp. 5425-5428, and U.S. 20130088726. In addition, spatial light modulators (SLM) are relative expensive, e.g., $1000 or more depending on the desired resolution, see Holoeye® or Texas Instruments DLP® products.
With S=1, a restricted version of the above approximation tolerant model is a good model for 2D imaging (one spatial and one depth dimension) with light pulses as the model minimizes a 1-D total variation of the depth map. However, to extend the model to 3-D volumes, i.e., signals sεNx×Ny×T (two spatial and one depth dimension) is unfortunately not obvious.